The Hopf algebra of Feynman graphs in QED

Abstract

We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT-identities, which leads in particular to the well-known identity $Z_1=Z_2$.

arXiv comment13 pages. Latex, uses feynmp. Minor corrections; to appear in LMP

References in corpus (8)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• An Étude in non-linear Dyson--Schwinger Equations
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• Quantum fields and motives
• Renormalization of QED with planar binary trees
• Using the Hopf Algebra Structure of QFT in Calculations
• Renormalization and motivic Galois theory

Cited by in corpus (20)

• Renormalization of gauge fields: A Hopf algebra approach
• Renormalized Quantum Yang-Mills Fields in Curved Spacetime
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Dyson Schwinger Equations: From Hopf algebras to Number Theory
• Recursion and growth estimates in renormalizable quantum field theory
• Gravity-Matter Feynman Rules for any Valence
• Feynman integrals and motives
• Diagrammatic Cancellations and the Gauge Dependence of QED
• Non-Linear Algebra and Bogolubov's Recursion
• Quantum field theory meets Hopf algebra
• Gauge Symmetries and Renormalization
• Log-Expansions from Combinatorial Dyson-Schwinger Equations
• Renormalization of gauge fields using Hopf algebras
• Hopf-algebraic Renormalization of QED in the linear covariant Gauge
• Supermanifolds from Feynman graphs
• Algebraic Structures in the Coupling of Gravity to Gauge Theories
• Algebra for quantum fields
• Geometrically relating momentum cut-off and dimensional regularization
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory